mateusdelimafilomeno

Texto

(1)Universidade Federal de Juiz de Fora Engenharia Elétrica Programa de Pós-Graduação em Engenharia Elétrica. Mateus de Lima Filomeno. Cooperative Communication for Broadband PLC and PLC/Wireless Systems: Achievable Data Rate Analyses. Juiz de Fora 2018.

(2) Ficha catalográfica elaborada através do Modelo Latex do CDC da UFJF com os dados fornecidos pelo(a) autor(a) Filomeno, Mateus. Cooperative Communication for Broadband PLC and PLC/Wireless Systems: Achievable Data Rate Analyses / Mateus de Lima Filomeno. – 2018. 62 f. : il. Orientador: Moisés Vidal Ribeiro Dissertação de Mestrado – Universidade Federal de Juiz de Fora, Engenharia Elétrica. Programa de Pós-Graduação em Engenharia Elétrica, 2018. 1. Comunicação via rede de energia elétrica. 2. Comunicação cooperativa. 3. Comunicação híbrida. 4. Comunicação sem fio. 5. Taxa de dados ergódica alcançável. I. Vidal Ribeiro, Moisés, orient. II. Título..

(3) Mateus de Lima Filomeno. Cooperative Communication for Broadband PLC and PLC/Wireless Systems: Achievable Data Rate Analyses. Dissertação de mestrado apresentada ao Programa de Pós-Graduação em Engenharia Elétrica da Universidade Federal de Juiz de Fora, na área de concentração em sistemas eletrônicos, como requisito parcial para obtenção do título de Mestre em Engenharia Elétrica.. Orientador: Moisés Vidal Ribeiro. Juiz de Fora 2018.

(4) I dedicate this thesis to my family and friends. A special feeling of gratitude to my mother Roseny, my grandmother Neide, and my brother Lucas..

(5) ACKNOWLEDGEMENTS First and foremost, I would like to thank God for guiding me throughout my life and for being my greatest refuge in difficult times. I would also like to thank to my mother Roseny de Lima Silva, my brother Lucas de Lima Filomeno, and my grandmother Neide de Araújo Lima, for always being by my side. My special words of thanks go to my friend Lucas Giroto de Oliveira, for years of friendship and revision of this text, and to my friends of the LCOM who never hesitated in helping me and made me grow professionally and personally. Thank you all, guys. Finally, my deepest thanks go to Professor Moisés Vidal Ribeiro, for his guidance throughout the development of this work, and to Professors Marcello Luiz Rodrigues de Campos and Fabrício Pablo Virgínio de Campos, for their valuable contributions..

(6) “I press on toward the goal to win the prize for which God has called me heavenward in Christ Jesus.” (Philippians 3:14).

(7) RESUMO Esta dissertação tem como objetivo discutir as comunicações cooperativas híbrida e não híbrida aplicadas a sistemas de comunicação de dados em banda larga e em ambientes residenciais. Nesse sentido, o modelo de canal com retransmissor único em dois estágios é investigado para sistemas de comunicação em banda larga através da rede de energia elétrica. Este modelo de canal cooperativo é formado pela concatenação de dois canais com retransmissor único, cobrindo enlaces de comunicação de dados com até dois saltos. Além disso, um modelo de canal híbrido com retransmissor único, utilizando rede elétrica e ar, é analisado para sistemas de comunicação de dados em banda larga em que enlaces de até um salto são considerados. Expressões de taxas de dados alcançáveis ergódicas são derivadas para os modelos de canais cooperativos híbridos e não híbridos, a fim de compará-los. Devido às características de canais e ruído das redes de energia elétrica, os resultados numéricos são baseados em um conjunto de dados constituído por estimativas de canais e medições de ruído cobrindo a faixa de frequência de 1, 7 a 100 MHz e diferentes posições do nó retransmissor. Para os canais sem fio, o modelo HIPERLAN/2 com a mesma largura de banda é utilizado, considerando uma freqüência central de 5 GHz, enquanto o ruído aditivo é considerado branco gaussiano. Com relação aos sistemas de comunicação através da rede energia elétrica, mostra-se que o modelo de canal com retransmissor único em dois estágios é a melhor opção quando o enlace da fonte ao destino encontra-se severamente degradado (por exemplo, alta atenuação de sinal devido à longa distância entre nós fonte e destino e/ou presença de ruído de alta potência). Quando a degradação do canal não é acentuada, o modelo de canal de dois saltos é mais apropriado. Acerca dos sistemas híbridos, constata-se que, quando o retransmissor está no meio do caminho entre fonte e destino, o modelo de canal híbrido com retransmissor único apresenta o melhor desempenho em termos de taxa de dados alcançável ergódica, enquanto o modelo de canal híbrido de um salto oferece os melhores resultados para outros casos. Palavras-chave: Comunicação cooperativa. Comunicação híbrida. Comunicação via rede elétrica. Comunicação sem fio. Taxa de dados ergódica alcançável..

(8) ABSTRACT This dissertation aims to discuss hybrid and non-hybrid cooperative communications applied to in-home broadband data communication systems. In this sense, the two-stage single-relay channel model is investigated for in-home broadband power line communication systems. This cooperative channel model consists of the concatenation of two single-relay channels, covering data communication links with up to two hops. Moreover, a hybrid power line/wireless single-relay channel model is analyzed for broadband data communication systems, considering one-hop links. Ergodic achievable data rate expressions are derived for both hybrid and non-hybrid cooperative channel models in order to compare them. Due to channel and noise characteristics of electric power grids, numerical results are based on a data set constituted by power line channel estimates and additive noise measurements covering the frequency band from 1.7 up to 100 MHz and different relay positions. For wireless channels, the HIPERLAN/2 model with the same bandwidth is used, but at a central frequency of 5 GHz, while the additive noise is considered to be Gaussian white. Regarding only power line communication systems, it is shown that the two-stage single-relay channel model is the best option when the source-to-destination link is severely degraded (e.g., high signal attenuation due to the long distance between source and destination nodes and/or high-power noise presence). When the channel degradation is not severe, the two-hop channel model is more appropriate. Concerning hybrid systems, it is observed that, when the relay is halfway between source and destination nodes, the hybrid single-relay channel model presents the best performance in terms of ergodic achievable data rate, while the hybrid one-hop channel model yields the best results for other cases. Key-words: cooperative communication, hybrid communication, power line communication, wireless communication, ergodic achievable data rate..

(9) LIST OF FIGURES Figure 1 – Two-stage single-relay channel model during one symbol time duration.. 17. Figure 2 – Adopted configurations from the 2S-SRC model. . . . . . . . . . . . . . 18 Figure 3 – Adopted hybrid channel models. . . . . . . . . . . . . . . . . . . . . . . 25 Figure 4 – PSD of an additive noise measurement in SD, SR, and RD channels. . 33. Figure 5 – β1 × β2 plane used to organize the data set in the four cases . . . . . . 34 Figure 6 – AF protocol and GSD = −80 dB: Ergodic achievable data rate gain vs. total transmission power. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 7 – DF protocol and GSD = −80 dB: Ergodic achievable data rate gain vs total transmission power. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 8 – AF protocol and ρB /ρE × GSD . . . . . . . . . . . . . . . . . . . . . . . 41 Figure 9 – DF protocol and ρB /ρE × GSD . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 10 – Without combination: Ergodic achievable data rate gain vs total transmission power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Figure 11 – With combination: Ergodic achievable data rate gain vs total transmis-. sion power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 12 – Magnitude of average PLC CFRs. . . . . . . . . . . . . . . . . . . . . . 60 Figure 13 – Magnitude of average wireless CFRs . . . . . . . . . . . . . . . . . . .. 61.

(10) LIST OF TABLES Table 1 – Decision matrices for MRC in the first and second time slots. . . . . . . 55 Table 2 – Decision matrices for MRC in the third and fourth time slots. . . . . . . 58 Table 3 – Specification of the 18-path HIPERLAN/2 for typical office environments. 59.

(11) ACRONYMS 2S-SRC two-stage single-relay channel AWGN additive white Gaussian noise AF amplify-and-forward CFR channel frequency response CIR channel impulse responses CSI channel state information DF decode-and-forward DFT discrete Fourier transform HIPERLAN high-performance radio local area network HS-OFDM hermitian-symmetric orthogonal frequency-division multiplexing HSRC hybrid single-relay channel HOHC hybrid one-hop channel LGRC linear Gaussian relay channel LTI linear time-invariant MIMO multiple-input multiple-output MRC maximum ratio combining N-CGRC N-block circular Gaussian relay channel PLC power line communication PSD power spectral density SNR signal-to-noise ratio SRC single-relay channel TDMA time-division multiple access.

(12) CONTENTS. 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.1 1.2. OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 THESIS ORGANIZATION . . . . . . . . . . . . . . . . . . . . . . . . . 15. 1.3. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 2. TWO-STAGE SINGLE-RELAY CHANNEL MODEL . . . . .. 2.1 2.2. PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . 16 ERGODIC ACHIEVABLE DATA RATE . . . . . . . . . . . . . . . . . 19. 2.2.1 2.2.2 2.2.3. Configuration A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Configuration B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Configurations C and D . . . . . . . . . . . . . . . . . . . . . . . . . 21. 2.2.4 2.3. Configuration E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. 3. HYBRID PLC/WIRELESS CHANNEL MODELS . . . . . . .. 3.1 3.2 3.2.1. PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . 25 ERGODIC ACHIEVABLE DATA RATE . . . . . . . . . . . . . . . . . 26 Hybrid One-Hop Channel Model . . . . . . . . . . . . . . . . . . . 27. 3.2.2 3.3. Hybrid Single-Relay Channel Model . . . . . . . . . . . . . . . . . 29 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 4 4.1. NUMERICAL RESULTS . . . . . . . . . . . . . . . . . . . . . . 33 TWO-STAGE SINGLE-RELAY CHANNEL MODEL . . . . . . . . . . 35. 4.1.1 4.1.2. Ergodic Achievable Data Rate vs Total Transmission Power . . 36 Ergodic Achievable Data Rate vs Average Channel Gain of the. 4.2 4.2.1. Direct Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 HYBRID PLC/WIRELESS COOPERATIVE CHANNEL MODELS . . 43 Ergodic Achievable Data Rate Without Combination . . . . . . 43. 4.2.2 4.3. Ergodic Achievable Data Rate With Combination . . . . . . . . 44 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. 5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.1. Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 16. 24. 48. 50.

(13) APPENDIX A – AF and DF achievable data rates . . . . . .. 55. APPENDIX B – SNR for configuration E . . . . . . . . . . . .. 57. APPENDIX C – Specification of the 18-path HIPERLAN/2 for typical office environments. . . . . . . . .. 59. APPENDIX D – PLC Channel Frequency Responses . . . . .. 60. APPENDIX E – Wireless Channel Frequency Responses . . .. 61. APPENDIX F – List of Publications . . . . . . . . . . . . . . .. 62.

(14) 12. 1 INTRODUCTION Currently, a challenging and timely research question is how to support the increasing demands of society and machines for connectivity that can fulfill previous and new kind of applications, such as broadband and narrowband communications with different trade-off among data rate, packet size, real or non-real time, energy availability, and spectrum scarcity. This research question leads to the emergence of novel energy-efficient, low-cost, flexible, and powerful data communication technologies and the improvement of existing ones. Among several technologies under investigation and improvement, power line communication (PLC) [1–7] and wireless communication [8–10] have received considerable attention due to their low-cost and suitability for fulfilling the needs and demands of Internet of Things, Smart Grids, and Smart Cities. While the former technology takes advantage of the ubiquitous presence of electric power grids [11, 12], the latter does not require infrastructures for connecting devices. Nonetheless, some challenges related to both technologies need to be pursued for maximizing their usage as data communication media. Regarding electric power grids, the main challenges are impedance mismatching, frequency selectivity of PLC channels, increasing signal attenuation along with both frequency and distance, high-power impulsive noise presence, dynamics of loads, electromagnetically unshielded power lines, and regulatory constraints [13–15]. Concerning wireless communication, besides high dependence on line-of-sight propagation and vulnerability to non-authorized access, it is worth mentioning the increasing signal attenuation along with both distance and carrier frequency, the susceptibility to interference among two or more systems operating in the same frequency band, the scarcity of spectrum, and the lack of reliable atmospheric conditions. In this context, cooperative communication [16, 17], which was initially developed to allow single-antenna devices to benefit from multiple-input multiple-output (MIMO) system features [10], has been widely investigated as one of the alternatives to counterbalance the disadvantages of data communication performed over electric power grids [18–21] and air [22–24]. In this regard, amplify-and-forward (AF) and decode-and-forward (DF) protocols have been the most reported in the literature [25–29]. Moreover, hybrid PLC/wireless systems have also been investigated as an alternative to improve data communication performance [30–32] since it can take advantage of both electric power grids and air to improve data rate and reliability between source and destination nodes under several circumstances, see [33]. In such data communication systems, electric power grids and wireless media are used simultaneously in a complementary way in order to maximize the available channel resources and fulfill data communication constraints. Regarding cooperative communication, an important issue is related to the cooperative channel model. Many models have been addressed in the literature, the most.

(15) 13 common being those with a single relay between edge nodes (the ones that are not relays). The simplest of them is called two-hop channel model, in which the relay is located between source and destination nodes and the direct link can be either existing [34] or non-existing [35]. Based on this model, [36] addressed a two-way channel model that is used in two phases: in the first one, the edge nodes perform data transmission towards the relay node, while in the second one, the relay node sends data towards the edges. Even more popular than the aforementioned models, the single-relay channel (SRC) model [20,28] considers that the relay is not exactly in between source and destination nodes, but rather connected by a branch point. With respect to the cooperative channel models with more than one relay, the multihop is the most investigated [18, 26]. It consists of multiple nodes between transmitter and receiver and considers that the direct link does not exist, i.e., each relay receives the signal only from its predecessor and forwards it to the subsequent relay until the signal reaches the destination node. Also, [37] addressed a cooperative channel model with several relays between source and destination nodes, proposing a method to find the best one to be used. On the subject of hybrid PLC/wireless systems, there are several works addressing this topic in the literature [30, 31, 33, 38–46]. Among them, different types of hybrid communication are shown. For instance, [38–40] focused on PLC and wireless technologies working in cascade so that the end-to-end data rate is bounded by the smallest one considering both channels. In such works, a node between source and destination is required, since there must be a device able to receive data at the output of the first channel and transmit through the other. More recently, [41] introduced a new concept of hybridism in which PLC and wireless channels are used in cascade without a intermediate node. Basically, transceivers are connected to electric power grids and to the air. They communicate with each other because both of them occupy the same frequency band starting in 1.7 MHz and ending in 100 MHz. According to [41], as the signal transmitted by a PLC device is irradiated from unshielded power lines, a wireless receiver and a PLC transmitter can be connected wirelessly. Another kind of hybridism associated with PLC and wireless communication refer to the use of these channels in parallel [33, 43–46], which proved to be able to improve data communication performance in terms of data rate [43, 46] and reliability [33, 44, 45]. Although some works have already pointed out the benefits of cooperative communication for PLC systems, a cooperative channel model resulting from the concatenation of two SRC models has not been discussed yet. Shortly, this channel model is characterized by using the destination node of a first SRC model as the source node of a second SRC model, resulting in a cooperative channel model with five nodes: source, destination, and three relays; which is named two-stage single-relay channel (2S-SRC) model. Using the same argument, the nS-SRC model (n > 2) can be easily defined; however, this work will drive attention over the 2S-SRC model because the use of up to two-hop data com-.

(16) 14 munication links makes sense when in-home broadband PLC systems are considered. As a matter of fact, data communication links with up to two hops within a home means that the distance between the edge nodes (source and destination nodes) is twice the one addressed in [20] and, as a consequence, the signal degradation between these nodes is much more severe. It means that it is worth investigating this scenario since up to two-hop links are feasible for performing in-home broadband data communication (i.e., frequency band is from 1.7 up to 100 MHz) due to the distances involved. Furthermore, [30,31,33] analyzed the narrowband hybrid PLC/wireless SRC model (i.e., a SRC model in which all nodes can perform data communication through both PLC and wireless channels) for dealing with low-bit-rate applications. In that case, a narrow frequency bandwidth of 500 kHz was considered. The frequency band used by PLC and wireless communication was between 0 and 500 kHz and between 915 MHz and 915.5 MHz, respectively. The adopted channel models showed that, for these frequency bands, PLC channels are frequency-selective, while the wireless channels are almost flat. Given some assumptions to ensure that only the channel diversity would be evaluated, it was shown that this kind of model can significantly improve the performance of a data communication system. Nonetheless, it is needed to verify whether this result is valid for a broadband hybrid PLC/wireless data communication system in which the frequency bandwidth is around 100 MHz. The broadband PLC is a baseband system that extends from 1.7 up to 100 MHz, while the broadband wireless communication is a passband system with carrier frequency at 5 GHz. As a consequence, both PLC and wireless channels are frequency-selective, which is different from the narrowband hybrid PLC/wireless investigations discussed in [30, 31]. 1.1 OBJECTIVES Given the aforementioned discussion and motivations related to cooperative and hybrid PLC/wireless data communication, the main objectives of this thesis are summarized as follows: • To investigate the usefulness of the 2S-SRC model for in-home broadband PLC systems by analyzing performances in terms of ergodic achievable data rate and considering different values of total transmission power, average channel gain of the direct link, and relative relays positions. Based on a data set from a measurement campaign, which is constituted by thousands of PLC channel estimates and several additive noise measurements, to provide concrete and realistic analyses that are suitable for PLC designers and practitioners involved with broadband data communication. • To analyze hybrid PLC/wireless channel models for broadband data communication.

(17) 15 within a home, deriving closed-form expressions for their ergodic achievable data rates and evaluating their performances for different values of total transmission power with the use of a measured data set of channels and additive noise for the PLC branch, and the high-performance radio local area network (HIPERLAN)/2 channel model [47] corrupted by additive white Gaussian noise (AWGN) for the wireless branch. 1.2 THESIS ORGANIZATION The remainder of this document is organized as follows: • Chapter 2 describes the 2S-SRC model for in-home broadband PLC systems. The most discussed cooperative channel models of the literature for PLC systems [28, 29, 34, 35], which are obtained from the 2S-SRC model, are also addressed. Closedform expressions for their ergodic achievable data rates are derived by considering AF and DF cooperative protocols. • Chapter 3 outlines hybrid channel models based on PLC and wireless communication systems. A formulation for the hybrid SRC model and the hybrid one-hop channel model is presented. Closed-form expressions for their ergodic achievable data rates are derived by considering the lack of signal combining as well as its presence. AF protocol is employed in this chapter. • Chapter 4 shows numerical analyses regarding the in-home broadband data communication systems described in Chapters 2 and 3. Considering different values of total transmission power, average channel gain of the direct link, and relative relays positions, comparative numerical analyses are carried out in terms of ergodic achievable data rate. • Chapter 5 places concluding remarks of this thesis. Also, it outlines future works. 1.3 SUMMARY This chapter has presented a brief introduction of this thesis, addressing important aspects of cooperative and hybrid communication applied to both PLC and wireless communication systems. Also, the main objectives and the organization of this work have been summarized..

(18) 16. 2 TWO-STAGE SINGLE-RELAY CHANNEL MODEL Several works have discussed the gains associated with the use of cooperative communication for improving the performance of in-home narrowband and broadband PLC systems [18, 20, 28]. The majority of them adopt well-known channel models in which PLC channels are random process [26, 28, 35]. A worthy remark regarding these PLC channel models is the fact that they consider frequency-domain representations in which the PLC channel is either flat or an stationary random process. Nevertheless, surveys on electric power grid measurements have shown that an extensive investigation has to be worldwide carried out to come up with representative PLC channel models. Trying to exploit this issue, [20] analyzed the cooperative communication adopting a data set constituted by several PLC channel estimates and additive noise measurements, which were obtained from a measurement campaign carried out in several Brazilian residences. Based on this data set, [20] discussed the suitability of the SRC model for in-home broadband data communication by addressing distances between source and destination nodes that cover up to one-hop links; however, the data set has shown that data communication covering up two-hop links has to be addressed since they cover distances in which the signal attenuation through in-home electric circuits is relevant. In this regard, this chapter investigates the 2S-SRC model and compares its ergodic achievable data rate against the ones from others cooperative channel models discussed in the literature for in-home broadband PLC systems. A formulation of the 2S-SRC model regarding in-home PLC system covering the frequency band from 1.7 up to 100 MHz is presented. Similar to [20], [48, 49] are used to derive the ergodic achievable data rates of five configurations of the 2S-SRC model as it allows to precisely take into account the frequency selectivity of PLC channels and the nonwhiteness of the additive noise in electric power grids. This chapter is organized as follows: Section 2.1 outlines the problem formulation, describing the cooperative channel models adopted in this chapter. In the sequel, Section 2.2 derives their ergodic achievable data rates, assuming AF or DF protocol at the relay node, as well as maximum ratio combining (MRC) technique to combine the signals at the destination node. Section 2.3 addresses a brief summary on this chapter. 2.1 PROBLEM FORMULATION The 2S-SRC model, shown in Fig. 1, is constituted by five nodes: one source node (S); three relay nodes (Ra , Rb , and Rc ); and one destination node (D). From this model, active (perform data transmission) and inactive (receive data) relays between source and destination nodes define eight distinct configurations. However, discussing all possible configurations would be rather confusing, as a large number of configurations and.

(19) 17 Ra hSRa [n]. Rc hRa Rb [n]. hRb Rc [n]. hRb D [n]. Rb. S hSRb [n]. D hRb D [n]. Figure 1: Two-stage single-relay channel model during one symbol time duration. active relays are possible. Then, this chapter focuses on the 2S-SRC model, considering configurations with at most one active relay besides the 2S-SRC itself, as they are the most studied in the literature. Therefore, the investigated configurations, which are shown in Fig. 2, are described as follows: • Configuration A: it is the one-hop channel model or the direct link. In this configuration all relays are inactive. • Configuration B: it is the two-hop channel model [35]. It assumes that Rb node is active, while Ra and Rc nodes are inactive.. • Configuration C: it is the SRC model [20]. Basically, Ra node is active, while Rb and Rc nodes are inactive. • Configuration D: it is the SRC model [20]. Essentially, Rc node is active, while Ra and Rb nodes are inactive. • Configuration E: it is the 2S-SRC model, in which all relays are active. To mathematically formulate the 2S-SRC model and its configurations, TC ≫ TS ,. where TC denotes the coherence time of the PLC channel and TS is the symbol time interval. Moreover, the time-division multiple access (TDMA) method is used to access. the channel since it is simple and very common in the literature [50, 51]. Then, each transmitter node (source and active relays) has one time slot from a total of NT time slots, with NT ∈ N∗ being defined as the number of transmitters in the analyzed configuration, to send information. Also, S and Rb nodes take advantage of the broadcasting characteristic of electric power grids; however, due to the high attenuation of some channels, Rc and D. nodes disregard all information received before Rb node performs its data transmission in the case Rb node is active (configurations B and E). For instance, in configuration E, S node broadcasts the original information to Ra and Rb nodes during the first time slot. In the following time slot, Ra node forwards the information to Rb node. In the third time slot, Rb node broadcasts the information to Rc and D nodes. At last, in the fourth time slot, Rc node sends the information to D node. Now, let the PLC channels be linear time-invariant (LTI) within a symbol time L. ij duration and the channel impulse responses (CIR) be represented by {hij [n]}n=1 , in which.

(20) 18 Conf. A. Conf. B. Conf. C. Conf. D. Conf. E. Figure 2: Adopted configurations from the 2S-SRC model. Lij is the length of the channel {hij [n]}, i ∈ {S, Ra , Rb , Rc } denotes the transmitter node. and j ∈ {Ra , Rb , Rc , D} denotes the receiver node. To be algebraically manipulated, the CIRs are represented as hij = [hij [1] hij [2] ... hij [Lij ]]T . Also, the N-length vector that represents the channel frequency response (CFR) associated with hij is expressed as . ILij. .  hij , Hij = WN  0(N −Lij )×N. (2.1). where WN ∈ CN ×N denotes the N -size discrete Fourier transform (DFT) matrix, IL ∈ RL×L denotes an L-size identity matrix and 0L×Q is a L × Q null matrix. Additionally, define the diagonal matrices ΛHij , diag{Hij [1], Hij [1], ..., Hij [N]} and Λ|Hij |2 , ΛHij Λ†Hij , in which Hij [k] is the k-th element of Hij , ∀ k ∈ {1, 2, ..., N}, and † denotes the conjugate transpose operator.. Let X ∈ CN ×1 and Vij ∈ CN ×1 be random vectors that represent, in the frequency domain, the symbol transmitted by S node and the additive noise at the output of the channel associated with i and j nodes. Next, given that E{·} denotes the expectation operator, it is assumed that E{X} = 01×N , RXX = E{XX† } = NIN , in which RMM represents the autocorrelation matrix of a finite-length random vector M. Also, E{Vij } = 0, and E{Vij ⊙ Vpq } = E{Vij } ⊙ E{Vpq }, ∀ij 6= pq, where ⊙ denotes the Hadamard product. It is also considered that E{Vij V†ij } = NΛPVij , in which ΛPVij , diag{PVij [1], PVij [2], ..., PVij [N]} and PVij [k] denotes the power of the k-th element of Vij , ∀ k ∈ {1, 2, ..., N}. Following, let PS ≥ 0, PRa ≥ 0, PRb ≥ 0, and PRc ≥ 0.

(21) 19 be the transmission powers allocated to S, Ra , Rb , and Rc nodes, respectively. The sum power constraint criterion is satisfied as follows: X i. Pi ≤ P,. (2.2). in which Pq ≥ 0q is the total transmission power. Next, Λ√Pi , q diag{ Pi [1], Pi [2], ..., Pi [N]} and ΛPi , diag{Pi [1], Pi[2], ..., Pi [N]}, where Pi [k] ≥ 0 is the power allocated to the k-th subchannel at i-th node, ∀ k ∈ {1, 2, ..., N}. Given the sum power constraint criterion and the TDMA method, the following research questions emerge: “What kind of improvement can the 2S-SRC model offer when the distance between edge nodes within a home corresponds to data communication links with up to two hops?”, “What kind of configuration can achieve the highest ergodic achievable data rate?” For answering these questions, Section 2.2 derives ergodic achievable data rate expressions for each configuration based on the aforementioned formulation. 2.2 ERGODIC ACHIEVABLE DATA RATE This section outlines expressions for the ergodic achievable data rate of the five aforementioned configurations using AF or DF at the relay node together with MRC at the destination node. To do so, every configuration is modeled as a linear Gaussian relay channel (LGRC) [48] with finite memory Lmax ∈ N|Lmax ≥ max{Lij }. However, in i,j PLC environment, noise is colored and CIRs have memory, which results in inter-block interference and, as a consequence, the evaluation of the achievable data rate for a LGRC model is very difficult. To overcome this problem, a recourse to N-block circular Gaussian relay channel (N-CGRC) [49], which eliminates the inter-block interference for N ≥ Lmax ,. is recommended. Note that the achievable data rate associated with LGRC tends to be equal to that for N-CGRC as N → ∞. Moreover, perfect symbol synchronization at the receiver side and complete channel state information (CSI) at both transmitter and receiver sides applies. The adoption of complete CSI allows the use of water-filling technique for maximizing the data rate associated with each channel model. 2.2.1 Configuration A Let YD = Λ√PS ΛHSD X + VSD. (2.3). be a complex random vector that models the received symbol at D node in the frequency domain. Also, X and VSD are Gaussian random vectors. According to [52], the mutual information between transmitted and received symbols is expressed as.

(22) 20. I(X; YD ) = h(YD ) − h(YD |X) = h(YD ) − h(Λ√PS ΛHSD X|X) − h(VSD |X) = h(YD ) − h(VSD ),. (2.4). where h(·) denotes the entropy operator over a random vector and h(Λ√PS ΛHSD X|X) = 0. In addition, the entropy of YD and VSD are, respectively, given by h(YD ) = log2 [(πe)N det(RYD YD )]. (2.5). h(VSD ) = log2 [(πe)N det(RVSD VSD )].. (2.6). and. Therefore, the mutual information is expressed as I(X; YD ) = log2 [det(IN + ΛγSD )],. (2.7). in which Λγij = ΛPi Λ|Hij |2 /ΛPVij is the signal-to-noise ratio (SNR) matrix associated. with the channel between i and j nodes.. As the mutual information is maximized when X is a Gaussian random vector, the ergodic achievable data rate for configuration A is given by .  . subject to. P i.  Bw log2 [det(IN + ΛγSD )] , CA = EH max   Λ N NT Pi. (2.8). Tr(ΛPi ) ≤ P , with i ∈ {S} and NT = 1. Note that EH {·} denotes the. expectation operator in relation to the CIRs that constitute the current configuration, Tr(·) denotes the trace operator, and Bw is the frequency bandwidth. 2.2.2 Configuration B In this configuration, the complex random vector YRb = Λ√PS ΛHSRb X + VSRb models the symbol received by Rb node in the frequency domain. Adopting AF, the received symbol at D node is expressed as YD = Λ√PR ΛPY ΛHRb D YRb + VRb D −1/2. Rb. b. = AX + BV,. (2.9). in which −1/2 A = Λ√PS Λ√PR ΛPY ΛHSRb ΛHRb D , Rb. b. −1/2. B = [Λ√PR ΛPY ΛHRb D b. Rb. IN ],. (2.10). (2.11).

(23) 21. V = [VTSRb VTRb D ]T ,. (2.12). and ΛPYR = ΛPS Λ|HSRb |2 + ΛPVSR is the power vector associated with YRb . As a result, b b the SNR matrix associated with the use of the AF protocol is given by. ΛγB,AF. , E{AX(AX)†}(E{BV(BV)†})−1. = ARXXA† (BRVV B† )−1 ΛPS ΛPRb Λ−1 PYR Λ|HSRb |2 Λ|HRb D |2 b = 2 ΛP + ΛPVR Λ ΛPRb Λ−1 |H PY VSR Rb D | Rb. b. (2.13). .. (2.14). bD. Similarly to configuration A, the mutual information is given by I(X; YD ) = log2 [det(IN + ΛγB,AF )].. (2.15). Consequently, the ergodic achievable data rate for configuration B using the AF protocol can be expressed as CB,AF subject to. P i. .  .  Bw log2 [det(IN + ΛγB,AF )] , = EH max   Λ N NT Pi. (2.16). Tr(ΛPi ) ≤ P , with i ∈ {S, Rb } and NT = 2.. For DF protocol, it is assumed that Rb node is error-free, which means that this node can corretly decode all information received from S node and, as a consequence, simplifies the evaluation of the achievable data rate. Therefore, considering the maximum flow-minimum cut theorem [53], the ergodic achievable data rate for configuration B adopting DF is given by CB,DF = EH where. subject to. . 1 min{CSRb , CRb D } , 2 . Bw log2 [det(IN + Λγij )], Cij = max ΛPi N NT P i. (2.17). (2.18). Tr(ΛPi ) ≤ P , with i ∈ {S, Ra , Rb , Rc }, j ∈ {Ra , Rb , Rc , D}, and NT = 1.. The term 21 in (2.17) is due to the presence of two transmitter nodes that use the channel during half of the available time interval (TDMA method). 2.2.3 Configurations C and D. Based on the fact that configurations C and D are SRC models [20], the functions fAF (·) and fDF (·) evaluate the achievable data rate of a SRC model using AF and DF, respectively (see Appendix A). Therefore, the ergodic achievable data rate for configurations.

(24) 22 C and D can be expressed as CC,AF = EH {fAF (ISD , ISRa , IRa D )} ,. (2.19). CC,DF = EH {fDF (ISD , ISRa , IRa D )} ,. (2.20). CD,AF = EH {fAF (ISD , ISRc , IRc D )} ,. (2.21). CD,DF = EH {fDF (ISD , ISRc , IRc D )} ,. (2.22). and. in which Iij may denote ΛPi , Λ|Hij |2 and ΛPVij , in accord with the chosen configuration and cooperative protocol. 2.2.4 Configuration E The mutual information between transmitted and received symbols in configuration E using AF is as follows: I(X; YD ) = log2 [det(IN + ΛγE,AF )],. (2.23). where ΛγE,AF is deduced in Appendix B. Thus, the ergodic achievable data rate for configuration E using AF is given by CE,AF subject to. P i.  . .  Bw = EH max log2 [det(IN + ΛγE,AF )] ,  Λ N NT  Pi. (2.24). Tr(ΛPi ) ≤ P , with i ∈ {S, Ra , Rb , Rc } and NT = 4.. Also, according to the maximum flow-minimum cut theorem and assuming that the relays are error-free, as in configuration B, the ergodic achievable data rate for configuration E using DF is expressed as CE,DF = EH. . 1 min{fDF (ISRb , ISRa , IRa Rb ), fDF (IRb D , IRb Rc , IRc D )} , 2 . (2.25). in which the term 21 is justified by the presence of two stages, as each one of them uses the channel during half of the available time interval. 2.3 SUMMARY This chapter has focused on the 2S-SRC model and some of the most discussed cooperative channel models in the literature for improving the PLC system performance. Thus, the problem formulation addressing five configurations has been presented as well as.

(25) 23 their ergodic achievable data rate expressions for both AF and DF cooperative protocols under the sum power constraint criterion and assuming the TDMA method to access the channel. MRC is used to signal combining..

(26) 24. 3 HYBRID PLC/WIRELESS CHANNEL MODELS The independent use of electric power grids and wireless media for data communication purpose has been pursued since long time ago. However, the need for maximizing their usage and fulfilling the astonishing and growing demands for connectivity among human beings and machines has brought attention to the drawbacks and limitations of these media. Attempting to address these issues, the investigation of the combined use of electric power grids and wireless media for data communication has started a few years ago. Currently, there is a great deal of attention in the parallel use of PLC and wireless channels to provide either reliability or high data rate, which has been called hybrid PLC/wireless data communication. There are several works exploring the hybrid PLC/wireless data communication, but this topic may be complicated since the behavior of electric power grids for narrowband (frequency band from 0 up to 500 kHz) and broadband (frequency band from 1.7 up to 100 MHz) are different in terms of signal propagation and additive noise influence. And the same is applied to the wireless medium. In this regard, [30, 31, 33] addressed the narrowband hybrid PLC/wireless data communication, while [43–45] discussed general results related to PLC and wireless channels models. Moreover, there is a lack of contribution regarding broadband hybrid PLC/wireless data communication in the literature. As well as investigations for narrowband hybrid PLC/wireless data communication, works related to broadband hybrid PLC/wireless data communication are not simple since the scope and coverage must be clearly defined. With respect to only the PLC side, it is important to pay attention to the voltage level (low-, medium- and high-voltage), the type of environment, such as indoor (vehicle, residences and building) and outdoor (metropolitan and rural areas), among other issues. Bringing the wireless side to the discussion, other issues have to be addresses, such as the frequency bands and distances between nodes. Based on the previous discussion, it is clear the need for correctly defining the frequency bands in which the hybrid PLC/wireless data communication is evaluated. In order to analyze the usefulness of the hybrid PLC/wireless data communication for broadband applications, this chapter focuses on hybrid channel models regarding a frequency band around 100 MHz. Driving the attention to one-hop links, only hybrid channel models in which the direct link is existing are analyzed in this chapter. By considering the frequency selectivity of both PLC and wireless channels, [48, 49] are used to derive the ergodic achievable data rates. This chapter is organized as follows: Section 3.1 addresses the problem formulation and outlines the hybrid PLC/wireless channel models adopted in this chapter. Section 3.2 derives the ergodic achievable data rates for these models and Section 3.3 presents a succinct summary about this chapter..

(27) 25 3.1 PROBLEM FORMULATION Two hybrid channel models for data communication purposes are adopted in this chapter (see Fig. 3). The first one does not consider the presence of active relays between S and D nodes and, as a consequence, it is named the hybrid one-hop channel (HOHC) model. This channel model was considered in previous works [43–46]. The second one is named the hybrid single-relay channel (HSRC) model and assumes that there is an active relay (R node) between S and D nodes, see details in [30, 31, 33]. In both models, all nodes can perform data communication over electric power grids and wireless media. In this work, the same symbols are transmitted through PLC and wireless channels by S and R nodes, which means that these channels are simultaneously accessed and the data communication aims to increase the reliability, coverage, and data rate. Based on the adoption of the TDMA method to access the channel, in the HSRC model, S node broadcasts the original information to R and D nodes, in the first time slot, while in the second one, R node forwards the information to D node. Let the symbol transmitted by S node, in the frequency domain, be a Gaussian random vector given by X ∈ CN ×1 , such that E{X} = 01×N , RXX = E{XX† } = NIN . Also, TCmax ≫ TS , where TCmax denotes the maximum coherence time considering both PLC and wireless channels and TS is the symbol time interval (duration). Moreover, both PLC and wireless channels are modeled as LTI within a symbol time interval and the vectors representing the CIR of these channels are expressed as hql = [hql [1] hql [2] ... hql [Lql ]]T , where Lql is the channel length, q ∈ {p, w} indicates the medium (power line or wireless), and l ∈. S. D. (a). R. S. D. (b). Figure 3: Adopted hybrid channel models. (a) HOHC model and (b) HSRC model..

(28) 26 {SD, SR, RD} denotes the links source-destination, source-relay, and relay-destination,. respectively. In addition, the N-length vector that represents the CFR of hql is Hql = WN [hql 01×N −l ]T and the diagonal matrices ΛqHl , diag{Hlq [1], Hlq [2], ..., Hlq [N]} and Λq|Hl |2 , diag{|Hlq [1]|2 , |Hlq [2]|2 , ..., |Hlq [N]|2 }, in which Hlq [k] represents the k-th element of Hql (k = 1, 2, ..., N).. Regarding the noise influence, let the additive noise at the output of the channel associated with the l link and q medium be represented by the Gaussian random vector Vql ∈ CN ×1 , in the frequency domain, such that E{Vql } = 0, E{Vql ⊙. Vqj } = E{Vql } ⊙ E{Vqj }, ∀ l 6= j and j ∈ {SD, SR, RD}. Also, E{Vpl ⊙ Vw j } = p q q w E{Vl } ⊙ E{Vj }, ∀j ∈ {SD, SR, RD} and E{Vl Vl †} = NΛPVq , in which ΛPVq , l l diag{PVql [1], PVql [2], ..., PVql [N]} and PVql [k] denotes the power of the k-th element of Vql . Next, let PSq ≥ 0 and PRq ≥ 0 be the transmission powers allocated to S and R nodes and q ∈ {p, w} medium, respectively. Similar to Chapter 2, the sum power constraint criterion is satisfied as follows:. X q. in which P q. q. ≥. (PSq + PRq ) ≤ P,. (3.1). 0 is the total transmission power.. diag{ PSq [1], PSq [2], ...,. Besides,. Λ√P q S. q. q. PSq [N]}, Λ√P q , diag{ PRq [1], R. q. PRq [2], ...,. ,. q. PRq [N]}, ΛPSq ,. diag{PSq [1], PSq [2], ..., PSq [N]}, and ΛPSq , diag{PRq [1], PRq [2], ..., PRq [N]}, where PSq [k] ≥ 0. and PRq [k] ≥ 0 are the power allocated to k-th subchannel and q medium at S and R nodes, respectively.. Based on the aforementioned formulation, the following questions arise: “Would a hybrid channel model benefit data communication between source and destination nodes when an in-home broadband data communication system is considered?”, “Which one of the aforementioned hybrid channel models can achieve the highest ergodic achievable data rate when representative CFR and additive noise are applied?” In order to answer these interesting questions, Section 3.2 formulates ergodic achievable data rate expressions for these models. 3.2 ERGODIC ACHIEVABLE DATA RATE This section aims to present the ergodic achievable data rate expressions for the adopted hybrid channel models. Such expressions will address the lack of combination as well as its presence at the destination node. The adopted roadmap aims to show the difference between both approaches when hybrid channel models are addressed. Regarding the HSRC model, the relay node adopts AF protocol and MRC technique to combine the received signals at the output of the PLC and wireless channels and the destination node also applies MRC technique for this purpose. This chapter only addresses the use of.

(29) 27 the AF protocol and the MRC technique because the former is quite considered as the typical cooperative protocol to be analyzed and the latter yields the optimal performance in terms of signal combining. Moreover, the additive noise is a circularly-symmetric Gaussian random vector. Note that the additive noise is white in the wireless side while it is colored in the PLC side. Furthermore, similar to Chapter 2 and [30, 31, 33], the use of LGRC [48, 49] is taken into account to deal with the inter-block interference created by PLC colored noise. Based on the fact that the adopted hybrid channel models have finite memory Lmax , in which Lmax ≥ max{Lql } and N → ∞, the achievable data rate of the adopted models can l,q. be approximated to that of N-CGRC.. 3.2.1 Hybrid One-Hop Channel Model First of all, let Yp and Yw be complex random vectors that represent the received signals at D node through power line and wireless media, respectively. Thus, the complex random vector that models the received symbol by D node is given by. Y′ =. . . Yp   Yw. . . Λ√P p ΛHpSD X + VpSD   S =   Λ√ w ΛHw X + Vw PS. . Λ√P p ΛHpSD . = . S. 0N ×N. = A′1 X′1 + V′1 , in which. . SD. SD. S. 0N ×N. T ′ T V′1 = [VpSD Vw SD ] , and X1 = [X X] .. . . . 0N ×N  X VpSD   +  w  Λ√P w ΛHwSD X VSD. Λ√P p ΛHpSD . A′1 = . . S. . 0N ×N  , Λ√ w ΛHw PS. (3.2). (3.3). SD. Without Combination: In this case, hardware complexity of D node is low and therefore it is not able to perform signal combining. It is assumed that the received signals from PLC and wireless channels are handled separately without considering a subsequent adoption of a combining technique (for more information on combining techniques, see [16]). Then, the resulting SNR matrix, without D node applying the combining technique, is expressed as.

(30) 28. †. ′ ′ ′ ′ † ′ ′ −1 Λw/o γHOHC = E{A1 X1 (A1 X1 ) }(E{V1 V1 }) †. = A′1 RX′1 X′1 A′1 (RV′1 V′1 )−1 . 0N ×N ΛP p Λ|HpSD |2 =  S 0N ×N ΛPSw Λ|HwSD |2 ΛP p Λ|Hp |2 S SD   ΛP p  V . =   . SD. 0N ×N. 0N ×N. −1.  ΛPVp  SD . 0N ×N   ΛPVw. 0N ×N. SD. .    . ΛPSw Λ|Hw |2  SD . (3.4). ΛPVw. SD. Note that the resulting SNR matrix is block diagonal and, as a consequence, it is composed of the SNR matrix of both media in the main diagonal. Therefore, the ergodic achievable data rate for the HOHC model is given by   . subject to. P q.   . Bw w/o CHOHC = EH max log2 [det(I2N + Λw/o γHOHC )] ,    ΛP q N NT S. (3.5). Tr(ΛPSq ) ≤ P , with q ∈ {p, w} and NT = 1.. With Combination: From Y′ , given by (3.2), and weight matrices, DqSD , the complex random vector that models the received symbol at D node in the frequency domain and after the use of the combining technique is given by Y =. i. h. ′ DpSD Dw SD Y. w = DpSD Yp + Dw SD Y w √ = (DpSD Λ√P p ΛHpSD + Dw )X + (DpSD VpSD + Dw SD Λ P w ΛHw SD VSD ) SD S. S. = A1 X + V1 ,. (3.6). w √ where A1 = DpSD Λ√P p ΛHpSD + Dw and V1 = DpSD VpSD + Dw SD Λ P w ΛHw SD VSD . As a SD S. S. consequence, the resulting SNR matrix after the use of the combining technique is given by † † −1 Λw/ γHOHC = E{A1 X(A1 X) }(E{V1 V1 }). = A1 RXX A1 † (RV1 V1 )−1 √ |2 |DpSD Λ√P p ΛHpSD + Dw SD Λ P w ΛHw SD S S . = 2 |DpSD |2 ΛPVp + |Dw SD | ΛPVw. (3.7). SD. SD. Assuming that the combining technique is MRC means that DqSD = Λ√P q ΛHq† /ΛPVq and, as a consequence, the SNR matrix for the HOHC model is given by Λw/ γHOHC =. ΛPSp Λ|HpSD |2 ΛPSw Λ|HwSD |2 + , ΛPVp ΛPVw SD. SD. S. SD. SD. (3.8).

(31) 29 which is the weighted sum of the SNR matrices associated with both media. As a result, the ergodic achievable data rate for the HOHC model with MRC is given by   . subject to. P q.   . Bw w/ CHOHC = EH max log2 [det(IN + Λw/ γHOHC )] ,  Λ q N NT   P S. (3.9). Tr(ΛPSq ) ≤ P , with q ∈ {p, w} and NT = 1.. 3.2.2 Hybrid Single-Relay Channel Model In this hybrid channel model, S and R nodes sends their information through both PLC and wireless channels. Therefore, in order to obtain the SNR matrix with and without combination, the vectorial representation of the four signals received by D node must be found. Two of them are originated in S node and described in Section 3.2.1, see (3.2), while the others came from R node. For describing the signals originated in R node, it is needed to use the random q vector YR = Λ√P q ΛHqSR X + VqSR that represents the symbol received by R node during S. the first time slot and through the q medium. Thus, with the possession of weight matrices DpSR and Dw SR , which are derived by the adoption of the MRC technique, the vector that represents the symbol at R node after the use of the combining technique is expressed as. YR =. h. Dw SR. A′2 X. V′2 ,. DpSR. i. . . Yp  R w YR. w √ = (DpSR Λ√P p ΛHpSR + Dw )X + (DpSR VpSR + Dw SR Λ P w ΛHw SR VSR ) SR S. S. =. +. (3.10). w √ and V′2 = DpSR VpSR + Dw where A′2 = DpSR Λ√P p ΛHpSR + Dw SR Λ P w ΛHw SR VSR . SR S. S. In the sequel, the transmitted symbol from R node to D node is represented by −1/2 † }/N due to the adoption of the AF protocol. XR = ΛPY YR , in which ΛPYR = E{YR YR R. Hence, the received symbols by D node that was originated in R node are given by.

(32) 30. . . Yp   Yw. . . Λ√P p ΛHpRD XR + VpRD   R =   Λ√ w ΛHw XR + Vw PR. RD. RD. . . −1/2 Λ√P p ΛHpRD [ΛPY (A′2 X + V′2 )] + VpRD   R R =   −1/2 (A′ X + V′ )] + Vw Λ√ w ΛHw [Λ PR. . PYR. RD. 2. −1/2 Λ√P p ΛPY ΛHpRD A′2 . = . 0N ×N. R. R. 0N ×N. . −1/2 Λ√P p ΛPY ΛHpRD V′2  R R  −1/2 Λ√P w ΛPY ΛHwRD V′2 R R. RD. 2. −1/2 Λ√P w ΛPY ΛHwRD A′2 R. R. . + VpRD  + Vw RD. . .  X . X. +. .. (3.11). As a result, the received symbol at D node considering the HSRC model is . Λ√P p ΛHpSD  S.        . Y′ =. 0N ×N. 0N ×N 0N ×N. 0N ×N. −1/2 Λ√P p ΛPY ΛHpRD A′2. 0N ×N. 0N ×N. 0N ×N Λ√ w ΛHw. 0N ×N. 0N ×N. 0N ×N. . R. R. PS. SD. .      X2     ′. +. −1/2 Λ√P w ΛPY ΛHwRD A2 R. R.  . + VpRD  . Λ√P w ΛPY ΛHwRD V2 + Vw RD R. R. 0N ×N. . VpSD.   √ −1/2 Λ ΛHpRD V′2 p ΛP YR PR    Vw  SD  −1/2 ′. +. 0N ×N. = A2 X2 + V2 ,.    . (3.12). in which . A2 =. Λ√P p ΛHpSD         . S. 0N ×N. 0N ×N. 0N ×N 0N ×N. 0N ×N. −1/2 Λ√P p ΛPY ΛHpRD A′2. 0N ×N. 0N ×N. 0N ×N Λ√ w ΛHw. 0N ×N. 0N ×N. 0N ×N. R. R. PS. . 0N ×N. SD. and X2 = [X X X X]T ..   .     ,     ′. −1/2 Λ√P w ΛPY ΛHwRD A2 R. R. (3.13). . VpSD    √ −1/2 p  p V′ + V Λ Λ p ΛP H RD  2 Y P  RD . V2 = . . R. R. ,.  Vw  SD  −1/2 ′ w √ w Λ P w ΛPY ΛHRD V2 + VRD R. R. (3.14). Without Combination: Once more, the SNR matrix of the received symbol without combination must be found, see (3.17) on page 32. Then, the ergodic achievable data rate.

(33) 31 for the HSRC model without combination is given by   . subject to. P i,q.   . Bw w/o CHSRC = EH max log2 [det(I4N + Λw/o γHSRC )] ,    ΛP q N NT i. (3.15). Tr(ΛPiq ) ≤ P , with i ∈ {S, R}, q ∈ {p, w}, and NT = 2.. With Combination: As in HOHC model, the SNR matrix for the HSRC model with combination, assuming that MRC is used, is equal to the sum of q media SNR matrices, see (3.18) on page 32. Therefore, the ergodic achievable data rate for the HSRC model after D node performs signal combining is given by   . subject to. P i,q.   . Bw w/ CHSRC = EH max log2 [det(IN + Λw/ γHSRC )] ,  N N  ΛP q  T i. (3.16). Tr(ΛPiq ) ≤ P , with i ∈ {S, R}, q ∈ {p, w}, and NT = 2.. 3.3 SUMMARY This chapter has concentrated on the HSRC and HOHC models. In this regard, it has presented the problem formulation related to these hybrid channel models and their ergodic achievable data rate expressions when the AF protocol applies together with or without signal combining at the destination node..

(34) † † −1 Λw/o γHSRC = E{A2 X2 (A2 X2 ) }(E{V2 V2 }). = A2 RX2 X2 A2 † (RV2 V2 )−1  ΛP p Λ|Hp |2 S SD  0N ×N  ΛP p  V SD   ΛP p ΛPYR Λ|Hp |2 |A′2 |2  0 R RD  N ×N ΛP p ΛPYR Λ|Hp |2 ΛPV′ +ΛPVp  2 RD R RD =     0N ×N 0N ×N     . 0N ×N. 0N ×N. 0N ×N. 0N ×N. 0N ×N. 0N ×N. ΛPSw Λ|Hw |2 SD ΛPVw. 0N ×N. 0N ×N. ΛPRw ΛPYR Λ|Hw |2 |A′2 |2 RD ΛPRw ΛPYR Λ|Hw |2 ΛPV′ +ΛPVw. SD. 2 where ΛPV′ 2 = |DpSR |2 ΛPVp + |Dw SR | ΛPVw .. RD. 2. RD.                 . (3.17). SR. SR. Λw/ γHSRC. . ΛPRp ΛPYR Λ|HpRD |2 |A′2 |2 ΛPSp Λ|HpSD |2 + = ΛPVp ΛPRp ΛPYR Λ|HpRD |2 ΛPV′ 2 + ΛPVp SD. RD. ΛPRw ΛPYR Λ|HwRD |2 |A′2 |2 ΛPSw Λ|HwSD |2 + . + ΛPVw ΛPRw ΛPYR Λ|HwRD |2 ΛPV′ 2 + ΛPVw SD. (3.18). RD. 32.

(35) 33. 4 NUMERICAL RESULTS In this chapter, performance analyses for the ergodic achievable data rates presented in Chapters 2 and 3 are carried out. For this purpose, a data set of more than 30, 000 estimates of in-home PLC channels and several additive noise measurements are used. This data set was obtained from a measurement campaign detailed in [19,20], which considered seven residences, covering houses and apartments (in-home facilities), located in the urban area of the city of Juiz de Fora, Brazil. This measurement campaign adopted the setup discussed in [6] and the methodology based on a hermitian-symmetric orthogonal frequency-division multiplexing (HS-OFDM) scheme reported in [54]. Also, it adopted a sampling frequency of 200 MHz; a frequency band (Bw ) from 1.7 to 100 MHz; an HS-OFDM symbol length of 2N = 4096; a symbol time interval of 23.04 µs; a frequency resolution of 48.8 kHz. Fig. 4 shows the noise power spectral density (PSD) of three of the additive noise measurements obtained from this measurement campaign.. PSD (dBm/Hz). -40. ij = SD ij = SR ij = RD. -60. -80. -100. -120 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. Frequency (MHz). Figure 4: PSD of an additive noise measurement in SD, SR, and RD channels. For wireless channel, the HIPERLAN/2 channel [55,56] is considered in this work. The HIPERLAN standard is a wireless local area network standard that has been defined by the European Telecommunications Standards Institute as an alternative to the IEEE 802.11 standards. The HIPERLAN second version [55, 56], also known as HIPERLAN/2, has been developed within a standardization project for broadband radio access networks. This version has five channel models that operates in 5 GHz band and can be deployed in several environments, such as offices, exhibition halls, and industrial buildings. Among them, the first one is chosen because it has been developed for typical office environments under the assumption of non-line-of-sight propagation conditions. Its specifications are presented in Appendix C. According to that, the delay spread is equal to 50 ns, resulting in a frequency bandwidth equal to 20 MHz. To obtain a wireless channel with a.

(36) 34 frequency bandwidth around to 100 MHz, this work considers a concatenation of five of these channels in the frequency domain. For analyzing the performance of the hybrid and non-hybrid cooperative channel models under different relays positions, the data set of measured PLC channels is divided into four cases. To do so, let hpSD , hpSR and hpRD denote CIRs of a generic SRC model, ||hp ||2 ||hpRD ||2 then the energy ratios β1 , ||hSR and β , (where || · || is the 2-norm of a vector) p 2 2 || ||hp ||2 SD. SD. make possible to organize the cases as follows:. • Case #1: the relay is located halfway between source and destination nodes, resulting in 0.4β1 ≤ β2 ≤ 2.5β1 and β12 + β22 ≥ 1. • Case #2: the relay is closer to the destination node than to the source node, resulting in 2.5β1 ≤ β2 and β12 + β22 ≥ 1. • Case #3: the relay is closer to the source node than to the destination node, resulting in 0.4β1 ≥ β2 and β12 + β22 ≥ 1. • Case #4: the relay is far from both source and destination nodes, resulting in β12 + β22 ≤ 1. Fig. 5 shows these cases, representing all measured PLC channels in a β1 × β2 plane in. which the dashed lines symbolize the separation among the cases. 24. Case Case Case Case. 20. #1 #2 #3 #4. 16. β2. 12. 8. 4. 0 0. 4. 8. 12. 16. 20. 24. β1 Figure 5: β1 × β2 plane used to organize the data set in the four cases covered in [20]..

(37) 35 The analyses presented in this chapter consider the total transmission power P ∈. {0, 10, 20, 30} dBm since these values cover ranges of transmission power that are allowed to data communication systems in which the considered channel models operate. 4.1 TWO-STAGE SINGLE-RELAY CHANNEL MODEL. In this section, the numerical results for the five configurations discussed in Chapter 2 are presented in terms of ergodic achievable data rate gain, which is defined as follows: ρα1 ,. Cα1 ,α2 , CA. (4.1). where α1 ∈ {A, B, C, D, E} refers to a configuration (see Fig. 2) and α2 ∈ {AF, DF } is the cooperative protocol. Note that CA ∈ {CA,AF , CA,DF } in accord with the type of the cooperative protocol and, as a consequence, the ergodic achievable data rate gain of configuration A is always equal to unity since it is taken as reference. For evaluating how the channel conditions can influence the behavior of the 2S-SRC model, this work shows analyses for. N 2 several average channel gain values of the HSD (GSD = 10 log10 (1/N k=1 |HSD [k]| )), while the PSD of the additive noise is always the same. This allow us to objectively. P. evaluate the advantages and disadvantages of cooperative protocols when GSD varies in a realistic range of values that are expected within a home. In order to calculate the ergodic achievable data rates, one hundred events are used for each configuration and case. For every event, a pair of SRCs of the same case is randomly selected within the data set. The first SRC composes the channels hSRb , hSRa , and hRa Rb , while the second one forms the channels hRb D , hRb Rc , and hRc D . To constitute the adopted configurations, the channels of these two SRC are concatenated. As an example, to obtain the CIR between Rb and D nodes (used in configuration C), simply concatenate hRa Rb and hRa Rb channels, in other words, hRa D = hRa Rb ⋆ hRb D , where ⋆ represents the convolution operator over two vectors. Note that both configurations A and B are not influenced by the relays locations, since the first does not have active relay nodes and the second has just Rb node which stays invariably halfway between S and D nodes. An insight on what happens in configurations C and D for different cases can be provided by analyzing configuration E under absence of Rc and Ra nodes, respectively. Concerning configuration E, case #1 keeps Ra node close to S and Rb nodes, while Rc node is close to both Rb and D nodes. Case #2 places Ra and Rc nodes close to S and Rb nodes, respectively, whereas case #3 places Ra and Rc nodes close to Rb and D nodes. At last, case #4 keeps both Ra and Rc nodes away from S, Rb , and D nodes..

(38) 36 4.1.1 Ergodic Achievable Data Rate vs Total Transmission Power As presented in [57], 90 % of PLC channel measurements in the frequency band from 1.7 to 100 MHz have average channel gain around −40 dB. As distances associated. with two-hop links in in-home facilities are considered, the ergodic achievable data rates are presented considering GSD = −80 dB, which is twice what is considered in [20], on average.. Fig. 6 shows the ergodic achievable data rate gains for the four analyzed cases when AF is adopted. Note that configuration B offers the best performance in terms of ergodic achievable data rate, always followed by configurations E, D, and C, in this order. It can also be noted that the gains are never greater than unity for configuration C and may vary significantly from case to case. For example, the ergodic achievable data rate gain for configuration B ranges from 2.1 up to 2.6 for case #1, while it ranges from 1.2 up to 1.8 for cases #2 and #3. If the set of P values covered higher values, it would be noticed that the ergodic achievable data rate gains tend to decrease along with P , which is not observed in Figs. 6b and 6c because the values of the adopted total transmission power are lower than 30 dBm. As considering such high P values would not be representative for practical systems, an initial instability in the curves causes intersections between them in Figs. 6b and 6c, masking the decreasing tendency of the achievable data rate gain along with P . Moreover, the greatest gains have shown up in case #1. For instance, configuration B yields gains as higher as 2.5 times the reference. In addition, one can see that the worst gains appear in case #4, since only configuration B yields gains greater than unity and, as a consequence, the others configurations are not useful for enhancing the data communication between source and destination nodes. Regarding cases #1 and #4, all configurations shows a tendency to decrease the gains as the total transmission power increases, while in other cases it will depend upon the configuration. Additionally, it is expected that as P tends to infinity, the gains achieved by configurations C and D tend to 1/2 since these configurations use the channel twice. Therefore, if the gain of one of these configurations is greater than unity for a given P and it starts to increase, then these gains decrease and, as a result, they become less than unity intersecting the reference. Because of this, ρD intersects ρA in Fig. 4a. Analyzing cases #2 and #3, it is noticed that configurations C and D achieve gains lower than unity, which means that it is not worth exploring the diversity of these configurations in such cases. It somehow disagrees with [20] because the current work addresses distances corresponding to two-hop links. On the other hand, by analyzing configurations B and E, one can see that the gains can surpass more than 1.3 and 1.7, respectively. Furthermore, these configurations show similar gains in cases #2 and #3, which occurs due to the existing mirroring between these two cases in these configurations..

(39) 37. α1 = A. ρα 1. α1 = B. α1 = C. 5. 5. 4.5. 4.5. 4. 4. 3.5. 3.5. 3. 3. ρα 1. 2.5. α1 = E. 2.5. 2. 2. 1.5. 1.5. 1. 1. 0.5. 0.5. 0. 0 0. ρα 1. α1 = D. 10. 20. 30. 0. 10. 20. P (dBm). P (dBm). (a). (b). 5. 5. 4.5. 4.5. 4. 4. 3.5. 3.5. 3. 3. ρα 1. 2.5. 2.5. 2. 2. 1.5. 1.5. 1. 1. 0.5. 0.5. 0. 30. 0 0. 10. 20. 30. 0. 10. 20. P (dBm). P (dBm). (c). (d). 30. Figure 6: AF protocol and GSD = −80 dB: Ergodic achievable data rate gain vs total transmission power. (a) case #1, (b) case #2, (c) case #3, and (d) case #4..

(40) 38 In fact, case #3 is obtained when S and D nodes positions changes in case #2 and vice versa. Fig. 7 shows the ergodic achievable data rate gains associated with DF. First of all, DF achieves higher gains than AF and the gains decrease inversely along with total transmission power, as expected. In comparison to AF, the performance ranking remains the same among the analyzed configurations and cases #1, #2, and #3. As an exception, this ranking is altered in case #1, where configuration E exceeds configuration B for low values of P . This results in an intersection between the performance curves for these configurations and is explained as follows. Assuming DF protocol, configuration E yields superior performance compared to configuration B for low P values due to the presence of more active relays between source and destination nodes. These relays can eliminate the noise effect before the signal is greatly attenuated and, as the total transmission power increases, their presence becomes unnecessary, resulting in ρB > ρE . Furthermore, as it occurs for AF, case #1 yields the best results while case #4 offers the worst ones. Focusing on case #1, one observes that the gains can reach high values mainly for configurations B and E. Besides, configuration C achieves significant gains, while configuration D does not achieve gains greater than unity. For case #4, configuration B is the only one that attains gains greater than unity. As the relays (represented by Ra and Rc nodes) are far from S and D nodes, configurations C, D, and E do not yield ergodic achievable data rate gains greater than unity for the considered values of P , as expected. Now, turning our attention to cases #2 and #3, it can be seen that configurations B and E have quite similar gains due to the mirroring between them as previously discussed. In addition, the gains are greater than unity for configurations B and E in these cases, while they are smaller than unity for configurations C and D. 4.1.2 Ergodic Achievable Data Rate vs Average Channel Gain of the Direct Link Overall, configurations B and E are the ones with the best performances in terms of ergodic achievable data rate. In addition, our results show that just for a specific situation (DF, case #1, P = 0 dBm) configuration E outperforms configuration B. However, it is our interest to evaluate whether this superiority occurs for different GSD values. For this reason, an analysis of ρB /ρE ratio versus GSD for distinct P values is carried out. If ρB /ρE > 1, then the ergodic achievable data rate achieved by configuration B is greater than the one achieved by configuration E, which means that it is better to use configuration B. On the other hand, if ρB /ρE < 1, then the opposite is valid. Based on the fact that the maximum and minimum average channel gain of in-home PLC channel are around −70 and -10 dB [57], respectively, and that two-hop links refer to.

(41) 39. α1 = A. ρα 1. α1 = B. α1 = C. 5. 5. 4.5. 4.5. 4. 4. 3.5. 3.5. 3. 3. ρα 1. 2.5. α1 = E. 2.5. 2. 2. 1.5. 1.5. 1. 1. 0.5. 0.5. 0. 0 0. ρα 1. α1 = D. 10. 20. 30. 0. 10. 20. P (dBm). P (dBm). (a). (b). 5. 5. 4.5. 4.5. 4. 4. 3.5. 3.5. 3. 3. ρα 1. 2.5. 2.5. 2. 2. 1.5. 1.5. 1. 1. 0.5. 0.5. 0. 30. 0 0. 10. 20. 30. 0. 10. 20. P (dBm). P (dBm). (c). (d). 30. Figure 7: DF protocol and GSD = −80 dB: Ergodic achievable data rate gain vs total transmission power. (a) case #1, (b) case #2, (c) case #3, and (d) case #4..

(42) 40 the concatenation of two PLC channels, this work considers GSD values from −180 to −20 dB, i.e., twice the range from [57] with an extra margin for the lower bound.. Fig. 8 shows ρB /ρE × GSD for distinct values of total transmission power when AF. is adopted. It can be seen that the ergodic achievable data rate of configuration B tends to be twice the same of configuration E, as the direct link conditions improve (GSD → 0), which is due to the number of time slots of these configurations. Disregarding case #4 (Fig. 8.d), if the total transmission power is low, ρE tends to be greater than ρB more quickly as GSD value decreases. To exemplify this statement, one can look at case #1 when P = 0 dBm and observe that configuration E is better than configuration B for GSD < −115 dB, while for P = 30 dBm this just happens for GSD < −175 dB. The same behavior is noticed in cases #2 and #3, but with a milder decrease in the value of ρB /ρE as GSD decreases. Although all curves start very close to GSD = −20 dB, the ρB /ρE ranges from 0.3 to 1.0 in case #1 and from 0.7 to 1.2 in cases #2 and #3 if GSD = −180 dB. A closer look at Fig. 8 shows that there is a range of GSD values in which ρB /ρE remains approximately constant. Such behavior means that the performance ratio between configurations E and B remains constant within this interval of GSD , i.e., the performances of configurations E and B vary at the same rate. It is important to say that, although the curves of different P apparently intersect each other, they portray distinct scenarios since they address different P values for a same noise PSD (see Fig. 4). Regarding case #4, one first observes that, as the GSD decreases, the value of ρB /ρE tends to be the same, unlike the previous cases. It means that no matter how much the average channel gain of the direct link decreases, configuration E is not better than configuration B, which confirms that the gains becomes less significant or even nonexistent, as the relays moves away from the direct link. In addition, such results clearly indicate how the use of TDMA method influences the ergodic achievable data rate gains. Due to the number of time slots, there is a reduction of 1/2 and 1/4 in the ergodic achievable data rates of configurations B and E, respectively. In conditions where there is no power allocated to Ra and Rc nodes, i.e., either direct link is not degraded or Ra and Rc nodes are far from the others, these reductions values are predominant in calculating the ρB /ρE ratio. Therefore, regardless the total transmission power, the ρB /ρE ratio is equal to 2 since configurations B and E use 2 and 4 time slots, respectively. Fig. 9 shows ρB /ρE considering DF. For cases #1, #2, and #3, if the total transmission power is low, then ρB /ρE decreases more quickly as GSD decreases, which is also observed if AF is adopted. Also, GSD values at which ρB /ρE = 1 for DF are lower than they are for AF. For example, regarding case #1 and P = 0 and 30 dBm, these values are, respectively, equal to −75 and −135 dB for DF, i.e., 40 dB greater than. what was observed for AF. By considering cases #2 and #3, one can observe that they.

(43) 41 2.5. P P P P. ρB /ρE. 2 1.5. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. -160. -180. GSD (dB). (a) 2.5. P P P P. ρB /ρE. 2 1.5. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. -160. -180. GSD (dB). (b) 2.5. P P P P. ρB /ρE. 2 1.5. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. -160. -180. GSD (dB). (c) 2.5. ρB /ρE. 2 1.5. P P P P. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. = 0 dBm = 10 dBm = 20 dBm = 30 dBm -160. -180. GSD (dB). (d). Figure 8: AF protocol and ρB /ρE × GSD . (a) case #1, (b) case #2, (c) case #3, and (d) case #4..

(44) 42 2.5. P P P P. ρB /ρE. 2 1.5. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. -160. -180. GSD (dB). (a) 2.5. P P P P. ρB /ρE. 2 1.5. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. -160. -180. GSD (dB). (b) 2.5. P P P P. ρB /ρE. 2 1.5. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 1 0.5 0 -20. -40. -60. -80. -100. -120. -140. -160. -180. GSD (dB). (c) 2.5. P P P P. 2. ρB /ρE. 10 1.5 1. 8. = 0 dBm = 10 dBm = 20 dBm = 30 dBm. 6 4. 0.5 0 -20. 2 -20. -40 -40. -60 -60. -80. -100 -120 -140 -160 -180 -80. -100. -120. -140. -160. -180. GSD (dB). (d). Figure 9: DF protocol and ρB /ρE × GSD . (a) case #1, (b) case #2, (c) case #3, and (d) case #4..